Titre :	Analytical and Numerical Treatment of Nonlinear Fractional Differential Equations Involving Caputo Fractional Operator
Type de document :	document électronique
Auteurs :	Lina Chetioui, Auteur ; Khalouta,Ali, Directeur de thèse
Editeur :	Setif:UFA
Année de publication :	2025
Importance :	1 vol (98 f.)
Format :	29 cm
Langues :	Anglais (eng)
Catégories :	Thèses & Mémoires:Mathématique
Mots-clés :	Fractional differential equations Caputo fractional derivative Approximate solution Analytical solution
Index. décimale :	510 - Mathématique
Résumé :	Nonlinear fractional differential equations play an important role in applied mathematics and physics. It is difficult to obtain the exact solution for these problems due to the complexity of the nonlinear terms included. In recent decades, there has been great development in the numerical analysis and exact solution for nonlinear fractional differential equations. The main objective of this thesis is to study the solutions of nonlinear fractional differential equations involving Caputo fractional operator by proposing new technique. To demonstrate the validity and reliability of this technique, it is applied to several numerical examples.
Note de contenu :	Sommaire Introductionii 1 Basicconceptsoffractionalcalculus1 1.1Applicationsoffractionalsystems........................ 1 1.1.1Automatic................................. 1 1.1.2Physics................................... 2 1.1.3Mechanicsofcontinuousmedia..................... 2 1.1.4Acoustic.................................. 3 1.2Functionalspaces................................. 3 1.2.1Spacesofintegrablefunctions...................... 3 1.2.2Spacesofcontinuousandabsolutelycontinuousfunctions....... 4 1.2.3Spacesofcontinuousfunctionswithweight............... 5 1.2.4BanachÂ…xedpointtheorem....................... 5 1.3SpeciÂ…cfunctionsforfractionalderivation................... 6 1.3.1Gammafunction............................. 6 1.3.2Betafunction............................... 7 1.3.3Mittag-Le­erfunction.......................... 7 1.4Fractionalintegralsandderivatives....................... 8 1.4.1FractionalintegralintheRiemann-Liouvillesense........... 8 1.4.2FractionalderivativeintheRiemann-Liouvillesense......... 12 1.4.3SomepropertiesoffractionalderivationinthesenseofRiemann- Liouville.................................. 15 1.4.4FractionalderivativeinthesenseofCaputo.............. 17 1.4.5SomepropertiesoffractionalderivationinthesenseofCaputo.... 22 1.4.6RelationbetweentheRiemann-LiouvilleapproachandthatofCaputo 23 2 Fractionaldi¤erentialequationsinthesenseofCaputo25 2.1EquivalenceresultbetweentheCauchyproblemandtheVolterraintegral equation...................................... 25 2.2Resultofexistenceanduniquenessofthesolution............... 27 3 Semi-analyticalmethodsandtheirconvergence33 3.1Adomiandecompositionmethod(ADM).................... 33 3.1.1Descriptionofthemethod........................ 33 3.1.2Adomianpolynomials........................... 35 3.1.3ConvergenceoftheADM......................... 36 3.2Homotopyperturbationmethod(HPM)..................... 39 3.2.1Descriptionofthemethod........................ 40 3.2.2Convergenceanalysis........................... 41 3.3Variationaliterationmethod(VIM)....................... 48 3.3.1Descriptionofthemethod........................ 49 3.3.2AlternativeapproachtoVIM...................... 49 3.3.3Convergenceanalysis........................... 51 3.4Newiterativemethod(NIM)........................... 56 3.4.1Descriptionofthemethod........................ 56 3.4.2ConvergenceofNIM........................... 57 4 Onthesolutionofnonlinearfractionaldi¤erentialequations61 4.1ApplicationoftheADM............................. 61 4.2ApplicationoftheHPM............................. 65 4.3ApplicationoftheVIM.............................. 69 4.4ApplicationoftheNIM.............................. 72 5 NewcombinationmethodforsolvingnonlinearfractionalLienardequa- tion 77 5.1Lienardequation................................. 77 5.2Khaloutatransform................................ 78 5.3Di¤erentialtransformmethod.......................... 82 5.4DescriptionoftheKHDTM........................... 84 5.5ConvergenceoftheKHDTM........................... 85 5.6Illustrativeexamples............................... 87 Conclusionandresearchperspectives92 Bibliography92
Côte titre :	DM/0205

Analytical and Numerical Treatment of Nonlinear Fractional Differential Equations Involving Caputo Fractional Operator [document électronique] / Lina Chetioui, Auteur ; Khalouta,Ali, Directeur de thèse . - [S.l.] : Setif:UFA, 2025 . - 1 vol (98 f.) ; 29 cm.
Langues : Anglais (eng)

Catégories :	Thèses & Mémoires:Mathématique
Mots-clés :	Fractional differential equations Caputo fractional derivative Approximate solution Analytical solution
Index. décimale :	510 - Mathématique
Résumé :	Nonlinear fractional differential equations play an important role in applied mathematics and physics. It is difficult to obtain the exact solution for these problems due to the complexity of the nonlinear terms included. In recent decades, there has been great development in the numerical analysis and exact solution for nonlinear fractional differential equations. The main objective of this thesis is to study the solutions of nonlinear fractional differential equations involving Caputo fractional operator by proposing new technique. To demonstrate the validity and reliability of this technique, it is applied to several numerical examples.
Note de contenu :	Sommaire Introductionii 1 Basicconceptsoffractionalcalculus1 1.1Applicationsoffractionalsystems........................ 1 1.1.1Automatic................................. 1 1.1.2Physics................................... 2 1.1.3Mechanicsofcontinuousmedia..................... 2 1.1.4Acoustic.................................. 3 1.2Functionalspaces................................. 3 1.2.1Spacesofintegrablefunctions...................... 3 1.2.2Spacesofcontinuousandabsolutelycontinuousfunctions....... 4 1.2.3Spacesofcontinuousfunctionswithweight............... 5 1.2.4BanachÂ…xedpointtheorem....................... 5 1.3SpeciÂ…cfunctionsforfractionalderivation................... 6 1.3.1Gammafunction............................. 6 1.3.2Betafunction............................... 7 1.3.3Mittag-Le­erfunction.......................... 7 1.4Fractionalintegralsandderivatives....................... 8 1.4.1FractionalintegralintheRiemann-Liouvillesense........... 8 1.4.2FractionalderivativeintheRiemann-Liouvillesense......... 12 1.4.3SomepropertiesoffractionalderivationinthesenseofRiemann- Liouville.................................. 15 1.4.4FractionalderivativeinthesenseofCaputo.............. 17 1.4.5SomepropertiesoffractionalderivationinthesenseofCaputo.... 22 1.4.6RelationbetweentheRiemann-LiouvilleapproachandthatofCaputo 23 2 Fractionaldi¤erentialequationsinthesenseofCaputo25 2.1EquivalenceresultbetweentheCauchyproblemandtheVolterraintegral equation...................................... 25 2.2Resultofexistenceanduniquenessofthesolution............... 27 3 Semi-analyticalmethodsandtheirconvergence33 3.1Adomiandecompositionmethod(ADM).................... 33 3.1.1Descriptionofthemethod........................ 33 3.1.2Adomianpolynomials........................... 35 3.1.3ConvergenceoftheADM......................... 36 3.2Homotopyperturbationmethod(HPM)..................... 39 3.2.1Descriptionofthemethod........................ 40 3.2.2Convergenceanalysis........................... 41 3.3Variationaliterationmethod(VIM)....................... 48 3.3.1Descriptionofthemethod........................ 49 3.3.2AlternativeapproachtoVIM...................... 49 3.3.3Convergenceanalysis........................... 51 3.4Newiterativemethod(NIM)........................... 56 3.4.1Descriptionofthemethod........................ 56 3.4.2ConvergenceofNIM........................... 57 4 Onthesolutionofnonlinearfractionaldi¤erentialequations61 4.1ApplicationoftheADM............................. 61 4.2ApplicationoftheHPM............................. 65 4.3ApplicationoftheVIM.............................. 69 4.4ApplicationoftheNIM.............................. 72 5 NewcombinationmethodforsolvingnonlinearfractionalLienardequa- tion 77 5.1Lienardequation................................. 77 5.2Khaloutatransform................................ 78 5.3Di¤erentialtransformmethod.......................... 82 5.4DescriptionoftheKHDTM........................... 84 5.5ConvergenceoftheKHDTM........................... 85 5.6Illustrativeexamples............................... 87 Conclusionandresearchperspectives92 Bibliography92
Côte titre :	DM/0205